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Download Special Relativity Part I Relativity Part I a.pdf RELATIVITY Einstein published two theories of relativity

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  • RELATIVITY

    Einstein published two theories of relativity

    In 1905

    The Special Theory

    For uniform motion 0a =

    In 1916

    The General Theory

    For non-uniform motion 0a ≠ .

  • 2

    First we will discuss The Special Theory

    At the beginning of the century most believed

    light was a wave.

    Thus must have something that waves:

    Sound has air

    Water has water, etc.

  • 3

    Physicists proposed that for waves of light

    something must wave.

    They called it the “ether” for light

    This ether then must fill the universe.

    The earth moves through the universe so:

    How fast are we traveling through the ether?

    To answer this we will start with a race

    between two boats (that run at exactly the same

    speed) in a river.

  • 4

    The boat going downstream will have speed

    c v+ The boat going upstream will have speed

    c v−

  • 5

    The calculated time for round trip is

    // 2 2

    2L L Lct c v c v c v

    = + = + − −

    The speed for the boat going cross-stream will

    have speed both ways

    2 2c v−

    Therefore

    2 2

    2 cross

    Lt v c

    = −

    Then the ratio of the two times will be

  • 6

    / / 2

    2

    1 1 1c r o s s

    t t v

    c

    = >

    Therefore we see that the boat that goes across

    and back wins the race even though both boats

    travel the same speed relative to the water.

    The main point is the time to complete the

    race is different for the two boats.

    Note:

  • 7

    We can measure ratio of times for boats and

    calculate the speed of river if we know the

    speed of boats.

  • 8

    The Michelson-Morley Experiment

    Diagram of Apparatus

  • 9

    Drawing of actual apparatus

    We know the speed of light.

    We want the speed of the earth through the

    ether.

  • 10

    When light arrives at the eye it has traveled two

    paths to reach the observer.

    There will be interference – either

    constructive

    or

    destructive

    The resulting image will be a series of lines.

  • 11

    Spectrometer lines

    Since the direction of the ether flow is not

    known the apparatus must be rotated.

  • 12

    First one than the other path will be parallel to

    the flow of ether.

    Therefore the interference lines should shift.

    Michelson and Morley did the experiment very

    carefully and did not find a shift.

    The conclusion has to be that:

    The Ether does not exist

    or

    the earth travels along with it.

  • 13

    Another experiment shows that we are

    not moving with it.

    Stellar Aberration is that experiment

    While the light travels down the telescope the

    telescope moves with the earth.

  • 14

    The telescope has to be tilted to keep the image

    in the center.

    If the ether (the substance that waves to cause

    the propagation of light) moves with the scope

    there would be no need to tilt the it.

    Therefore we conclude the ether does not

    exist.

    Classical Relativity

    The transformation equations before Einstein

  • 15

    '

    '

    '

    '

    x x v t

    y y

    z z

    t t

    = −

    =

    =

    =

    These are the Galilean Equations that allow

    observers to compare observations in two

    different frames moving relative to each other

    with constant velocity.

  • 16

    Observer on ground and observer on railroad

    car moving in x-direction.

    The observer on the ground observes the birds

    separated by distance 2 1x x− .

    The distances are equal.

  • 17

    If an airplane flies over the railroad car

    traveling in the x+ direction at a speed xu

    measured by the observer on the ground what

    will be the speed ( 'xu ) of the airplane measured

    by the observer on the railroad car?

    We can use the transformation equation for x

    'x x vt= − and the equation for t

    't t=

    Differentiate and divide to get

  • 18

    'dx dx dvv t

    dt dt dt = − −

    ' 0x xu u v= − −

    if the velocity of the railroad car is constant.

    If the observer on the ground measures the

    velocity of the airplane as

    xu

    then the person on the railroad car will measure

    ' xu

  • 19

    What if the person on the ground points a

    flashlight in the x+ direction? What will be

    the speed of light measured by the observer on

    the railroad car?

  • 20

    We get

    '

    '

    x xu u v giving c c v

    = −

    = −

    We must keep this result in mind as we discuss

    Einstein’s Theory.

  • 21

    Einstein’s postulates for the Special Theory of

    Relativity:

    1. Fundamental laws of physics are identical

    for any two observers in uniform relative

    motion.

    2. The speed of light is independent of the

    motion of the light source or observer.

  • 22

    These postulates cannot be satisfied using the

    Galilean Equations, as we will see.

    However Einstein found that the following

    equations worked.

    2

    2

    2

    ' ( ) ' '

    ' ( )

    1

    1

    x x vt y y z z

    vxt t c

    where

    v c

    γ

    γ

    γ

    = − = =

    = −

    =

  • 23

    These are the Lorentz Transformation

    Equations.

    Now consider the airplane flying over the

    railroad car in the x-direction. What is the

    speed of the airplane as measured by the

    observer on the ground? What is the speed of

    the airplane as measured by the observer on the

    railroad car? We need to answer these

    questions by using the Einstein-Lorenze

    Equations.

  • 24

    2

    2

    2

    ' ( )

    ' ( )

    ' ( )

    ' ( )

    ' '

    x x vt and

    vxt t c

    differentiate dx dx vdt and

    vdxdt dt c

    divide dx dx vdt

    vdxdt dt c

    γ

    γ

    γ

    γ

    = −

    = −

    = −

    = −

    − =

  • 25

    divide by 'dt

    2

    '

    2

    ' '

    1

    1

    x x

    x

    dx vdx dt dxdt v dt

    c or

    u vu vu c

    − =

    − =

    Thus if an object (an airplane) flies over the

    railroad car the observer on the ground will

  • 26

    measure the speed in the x direction as xu . The

    observer on the car will find 'xu .

    What about the speed of light when a flashlight

    is pointed in the x-direction?

    The observer on the ground points a flashlight

    in the +x direction. What will be the speed of

    light measured by the observer on the car?

    '

    2 ( ) /1 1

    x c v c v c vu cvc v c v c

    c c

    − − − = = = =

    −− −

  • 27

    Both observers, even though they are moving

    relative to each other, measure the same value

    for the speed of light.

    This is in agreement with the Second Postulate.

    LENGTH CONTRACTION

    Read the section on Length Contraction in the

    book. We will do it a little differently.

  • 28

    The observer on the moving railroad car has a

    rod moving with him. He measures the length

    of the rod to be

    ' ' 2 1 0x x L− =

    Use the Lorentz equations to get

    ' 2 2 2

    ' 1 1 1

    ( )

    ( )

    x x vt

    x x vt

    γ

    γ

    = −

    = −

    Then putting these in the equation

  • 29

    [ ]

    0 2 2 1 1

    0 2 1 2 1

    ( ) ( )

    ( ) ( )

    L x vt x vt

    L x x v t t

    γ γ

    γ

    = − − −

    = − − −

    If the observer on the ground measures the far

    end and near end of the rod at the same time

    1 2t t= Then

    0 2 1( )L x x Lγ γ= − =

    or

    0LL γ

    = and γ > 1

  • 30

    So the observer on the ground with the rod

    moving past in the x direction measures the rod

    to be shorter than what is measured by the

    observer at rest relative to the rod and on the

    car.

    Length Contraction is a prediction of the

    Lorentz Equations.

    TIME DILATION

    Again we will find time dilation a different way

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